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\begin{equation}
{\bm \Psi}^{\prime}(t)=\left[{\bf x}^{\prime (1)}, {\bf x}^{\prime (2)},\cdots, {\bf x}^{\prime (n)}\right]
=\left[{\bf P}(t)\, {\bf x}^{(1)}, {\bf P}(t)\, {\bf x}^{(2)}, \cdots, {\bf P}(t)\, {\bf x}^{(n)}\right]
={\bf P}(t) \left[{\bf x}^{(1)}, {\bf x}^{(2)}, \cdots, {\bf x}^{(n)}\right]
={\bf P}(t) \, {\bm \Psi}(t).\tag{6.6.5}
\end{equation}
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